\newproblem{lay:4_2_11}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.2.11}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	For the set below, either find an appropriate theorem to show that $W$ is a vector space or find a specific example to show the contrary.
	\begin{center}
		$W=\left\{\begin{pmatrix}s-2t\\3+3s\\3s+t\\2s\end{pmatrix}|\forall s,t\in\mathbb{R}\right\}$
	\end{center}
}{
  % Solution
	$W$ is not a subspace because $\mathbb{R}^4\ni \mathbf{0}\notin W$. To show why, consider the vector equation
	\begin{center}
		$\begin{pmatrix}s-2t\\3+3s\\3s+t\\2s\end{pmatrix}=\begin{pmatrix}0\\0\\0\\0\end{pmatrix}$
	\end{center}
	The equation for the last component implies
	\begin{center}
	   $2s=0\Rightarrow s=0$
	\end{center}
	But for the second component
	\begin{center}
	   $3+3s=0\Rightarrow s=-1$
	\end{center}
	Since $s$ cannot take the values 0 and -1 at the same time, we conclude that there are no values of $s$ and $t$ such that $\mathbf{0}\in W$, and
	consequently, the set $W$ cannot be a vector space.
}
\useproblem{lay:4_2_11}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
